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added to multivector field further references on how the divergenc/BV operator is the dual of the de Rham differential.
Domenico, could you tell me if you think that the following statement is correct?
in full abstractness, the content of Lagrangian BV is this:
we
start with a configuration space of sorts
and assume we have a fixed isomorphism between its Hochschild cohomology and Hochshcild homology, which we think of as an iso between its differential forms and its multivector fields induced by a volume form (which it is for finite dimensional spaces);
then we think of an action functional times a volume form on our configuration space as a closed differential form $\exp(i S) vol$, hence as an element in the Hochschild homology that is also in the cyclic homology
and then use the above isomorphism to think of this equivalently an element in Hochschild cohomology, being a cocycle in cyclic cohomology.
the cyclic differential is the BV-operator and the closure condition is the “master equation” $\Delta \exp(i S) = 0$;
the fact that Lagrangian BV is controled by BV-algebra and hence, by Getzler’s theorem, by algebra over the homology of the little framed disk operad now follows from the fact that Hochschild homology of our space is given by the derived loop space.
Is that right? Is that the NiceStoryAboutLagrangianBV™? If so, is this written out in this fashion explicitly somewhere?
Hi Urs,
yes I think that’s right. A closely related reference is Batalin-Vilkovisky algebra structures on Hochschild cohomology, by Luc Menichi
yes I think that’s right.
Great. Thanks. Now that makes a huge amount of previously disconnected structure (in my mind) form a single coherent whole. Derived loop spaces rule supreme.
I’ll check out the references that you provided, thanks.
Wait a moment, I think I need more information:
I know that there is plenty of discussion of BV-algebra structure on Hochschild cohomology.
What I am looking for is explicit discussion of the connection to the discussion of master equations for BV action functionals. By the above chain of thoughtsit should be clear how both are related. But I have trouble finding a reference that explicitly says this.
The references that I can see all seem to either discuss BV-structure on Hochschild cohomology abstractly, or discuss BV-action functionals, but not both. Your article was a pleasant exception!
I have added to BV-algebra a reference by C. Roger that comes pretty close to what I am looking for, but it’s still not quite it. What I would like to see is the explicit statement (if anyone has made that!) that restates Witten’s old observation in Hochschild- and then in derived loop space terms:
Witten observed that the BV-action functional is just the action functoional times a hypothetical “mid-dimensional volume form” regarded under the isomorphism of differential forms with multivector fields (on an oriented space), and that the master equation therefore just says dually that $\exp(i S) vol$ is closed.
Now we should go two steps further:
realize that the duality between forms and multivector fields is a special case of duality between Hochschild homology and cohomology;
realize that these are cohomology and homology on derived loop spaces, with the differential coming from the loop rotation.
Because combined with Witten’s old observation and Getzler’s theorem this gives the geometric explanation for why there is BV-algebra in lagrangian quantum field theory.
I have added to Poincare duality references about Poincare duality on Hochschild (co)homology.
I think that is what’s going on abstractly: Lagrangian BV is the image under Hochschild-Poincare duality of ordianary integration theory of differential forms regarded as living in Hochschild homology.
Or rather, and I guess this is the point: in the absence of Poincare duality (due to infinite-dimensionality of our space) we may nevertheless behave as if the Hochschild cohomology with its BV structure is dual to integration theory of forms, thereby generalizing integration theory of forms to infinite-dimensional situations.
Something like this.
let me see if I’m following what you’re after. you’re saying: on a finite dimensional manifold I have a nice classical theory of differential forms, de Rham differential and integration; then we can do a quite weird thing: pick up a volume form and use it to make an image of this pretty nice theory in terms of multivector fields. we obtain a bizzarre structure, which goes under the name of BV-algebra structure. then you say: wait, you’re wrong in saying that this is bizzarre, this is actually quite natural and comes from the geometry of (derived) loop space, which is a very general and solid construction: it is the classical theory of integration of differential forms to be something peculiar. and indeed, in the infinite dimensiona case I’m still able to talk of the loop space construction, whereas I’m not able anymore to give a meaning to classical integration. so, you see, intergration is really a BV-thing by its very nature.
or something like this :)
yes, something like this.
Maybe another way to ask the question that I meant to ask is the following:
given an action functional $\exp(i S) : C \to U(1)$ it just so happens that
the graded algebra of multivector fields equipped with the differential $[S,-]_{Schouten}$ is a derived resolution of the critical locus (at last if we are lucky and the critical locus is not too wild);
the equation $\Delta \exp(i S) = 0$ in that BV-algebra of multivectorfields, regarded as Hochschild cohomology, expresses, under would-be Poincare-Hochschild cohomology, the fact that $\exp(i S) vol$ is Connes-coboundary closed.
Is this a coincidence that Hochschild cohomology = multivector fields plays these two different but both crucial roles in Lagrangian BV?
The second one, in any case, explains geometrically (via derived loop spaces) why the BV-quantization of any system whatsoever should be related to the 2d TQFTas seen by the framed little disk operad.
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